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Physical review. B, Condensed matter

Spectrum of the correlation matrix in short-range Ising spin glasses

Anil Khurana

Published Nov 1, 1989• 1 citation
Publisher

Abstract

In the droplet picture of the scaling theory for the short-range Ising spin glass, the largest eigenvalue of the correlation matrix diverges algebraically with the Edwards-Anderson susceptibility, the exponent being d/(d-\ensuremath{\theta}), where \ensuremath{\theta} is the stiffness exponent. The spectral density for the correlation matrix has a bump near the lower edge of the spectrum. Comparing these predictions with results from numerical studies of the spectrum might provide a unique test of the droplet picture.

DOI
10.1103/physrevb.40.9279

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In the droplet picture of the scaling theory for the short-range Ising spin glass, the largest eigenvalue of the correlation matrix diverges algebraically with the Edwards-Anderson susceptibility, the exponent being d/(d-\ensuremath{\theta}), where \ensuremath{\theta} is the stiffness exponent. The spectral density for the correlation matrix has a bump near the lower edge of the spectrum. Comparing these predictions with results from numerical studies of the spectrum might provide a unique test of the droplet picture.
We propose a new picture of the Ising-spin-glass phase, based on an Ansatz for the scaling of low-lying large-scale-droplet excitations. We find behavior very different from the infinite-range model. The truncated spatial correlations decay as a power of distance, the ac nonlinear susceptibility diverges as a power of $\mathrm{ln}\ensuremath{\omega}$, and the magnetization noise power diverges as $\frac{1}{\ensuremath{\omega}}$ with logarithmic corrections. A magnetic field destroys the spin-glass phase so that there is no de Almeida-Thouless transition. Defect excitations should yield similar dynamic phenomena in vector spin-glasses.
Domain-wall free energy $\ensuremath{\delta}F$, entropy $\ensuremath{\delta}S$, and the correlation function Ctemp{C}_{\mathrm{temp}}Ctemp​ of $\ensuremath{\delta}F$ are measured independently in the four-dimensional $\ifmmode\pm\else\textpm\fi{}J$ Edwards-Anderson (EA) Ising spin glass. The stiffness exponent $\ensuremath{\theta}$, the fractal dimension of domain walls ds{d}_{s}ds​, and the chaos exponent $\ensuremath{\zeta}$ are extracted from the finite-size scaling analysis of $\ensuremath{\delta}F$, $\ensuremath{\delta}S$, and Ctemp{C}_{\mathrm{temp}}Ctemp​, respectively, well inside the spin-glass phase. The three exponents are confirmed to satisfy the scaling relation $\ensuremath{\zeta}={d}_{s}/2\ensuremath{-}\ensuremath{\theta}$ derived by the droplet theory within our numerical accuracy. We also study bond chaos induced by random variation of bonds, and find that the bond and temperature perturbations yield the universal chaos effects described by a common scaling function and the chaos exponent. These results strongly support the appropriateness of the droplet theory for the description of chaos effect in the EA Ising spin glasses.
In this Letter we address the nature of broken ergodicity in the low temperature phase of Ising spin glasses by examining spectral properties of spin correlation functions C(ij) identical with<S(i)S(j)>. We argue that more than one extensive [i.e., O(N)] eigenvalue in this matrix signals replica symmetry breaking. Monte Carlo simulations of the infinite-range Ising spin-glass model, above and below the Almeida-Thouless line, support this conclusion. Exchange Monte Carlo simulations for the short-range model in four dimensions find a single extensive eigenvalue and a large subdominant eigenvalue consistent with droplet model expectations.
We investigate the chaotic temperature dependence of the renormalized couplings in a short-range Ising spin glass. Some of our analytic and numerical results, obtained by the Migdal-Kadanoff renormalization scheme but argued to have general validity, differ from those of the scaling and droplet theory. First, chaos is present also at and above the critical temperature. Second, between dl{\mathit{d}}_{\mathit{l}}dl​=2.46 (the lower critical dimension) and d+{\mathit{d}}_{+}d+​=3.4, the chaos in the critical region is characterized by a new critical exponent ${\mathrm{\ensuremath{\zeta}}}_{\mathit{c}}$ that has hitherto escaped attention.
We study the nature of the broken ergodicity in the low temperature phase of Ising spin glass systems, using as a diagnostic tool the spectrum of eigenvalues of the spin-spin correlation function. We show that multiple extensive eigenvalues of the correlation matrix ${C}_{\mathrm{ij}}\ensuremath{\equiv}〈{S}_{i}{S}_{j}〉$ occur if and only if there is replica symmetry breaking. We support our arguments with Exchange Monte Carlo results for the infinite-range problem. Here we find multiple extensive eigenvalues in the replica symmetry breaking (RSB) phase for $N\ensuremath{\gtrsim}200,$ but only a single extensive eigenvalue for phases with long-range order but no RSB. Numerical results for the short-range model in four spatial dimensions, for N< 1296,N<~1296,N< 1296, are consistent with the presence of a single extensive eigenvalue, with the subdominant eigenvalue behaving in agreement with expectations derived from the droplet model. Because of the small system sizes we cannot exclude the possibility of replica symmetry breaking with finite size corrections in this regime.